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Theorem conncompss 23442
Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompss ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑋)
2 conntop 23426 . . . . . . 7 ((𝐽t 𝑇) ∈ Conn → (𝐽t 𝑇) ∈ Top)
323ad2ant3 1135 . . . . . 6 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐽t 𝑇) ∈ Top)
4 restrcl 23166 . . . . . . 7 ((𝐽t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
54simprd 495 . . . . . 6 ((𝐽t 𝑇) ∈ Top → 𝑇 ∈ V)
6 elpwg 4602 . . . . . 6 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
73, 5, 63syl 18 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
81, 7mpbird 257 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9 3simpc 1150 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn))
10 eleq2 2829 . . . . . 6 (𝑦 = 𝑇 → (𝐴𝑦𝐴𝑇))
11 oveq2 7440 . . . . . . 7 (𝑦 = 𝑇 → (𝐽t 𝑦) = (𝐽t 𝑇))
1211eleq1d 2825 . . . . . 6 (𝑦 = 𝑇 → ((𝐽t 𝑦) ∈ Conn ↔ (𝐽t 𝑇) ∈ Conn))
1310, 12anbi12d 632 . . . . 5 (𝑦 = 𝑇 → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) ↔ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
14 eleq2 2829 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
15 oveq2 7440 . . . . . . . 8 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
1615eleq1d 2825 . . . . . . 7 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
1714, 16anbi12d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1817cbvrabv 3446 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)}
1913, 18elrab2 3694 . . . 4 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
208, 9, 19sylanbrc 583 . . 3 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
21 elssuni 4936 . . 3 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
2220, 21syl 17 . 2 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
23 conncomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2422, 23sseqtrrdi 4024 1 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  {crab 3435  Vcvv 3479  wss 3950  𝒫 cpw 4599   cuni 4906  (class class class)co 7432  t crest 17466  Topctop 22900  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-rest 17468  df-top 22901  df-conn 23421
This theorem is referenced by:  conncompcld  23443  tgpconncompeqg  24121  tgpconncomp  24122
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